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Fatigue damage modeling in solder interconnects using a cohesive zone approach
1. Fatigue damage modeling in solder interconnects
using a cohesive zone approach
Adnan Abdul-Baqi, Piet Schreurs, Marc Geers
AIO-Meeting: 03-06-2003
Supported by Philips
2. Outline
โข Introduction
โข Geometry and loading
โข Cohesive zone method:
โ Cohesive zone formulation
โ Cohesive tractions
โ Damage evolution law
โ One dimensional example
โข Results:
โ Damage distribution
โ Corresponding total effective damage and reaction force
โ Life-time prediction in comparison with empirical models
โข Conclusions
3. Printed circuit board (PCB)
โข Solder joints provide mechanical & electrical connection between the silicon
chip and the printed circuit board.
โข Repeated switching of the device โ temperature ๏ฌuctuations โ fatigue of the
solder joints โ device failure.
5. Tin-Lead solder
Typical Tin-Lead microstructure (A. Matin).
โข Simpli๏ฌed microstructure is chosen for the simulations:
โ Physically: rapid coarsening โ continuous change.
โ Numerically: Large number of degrees of freedom โ time consuming.
6. Geometry and loading: solder bump
Ux
0.1 mm
Lead
y
Tin
x 0.1 mm
โข Plane strain formulation, thickness = 1 mm.
โข Elastic properties: Tin (E = 50 GPa, ฮฝ = 0.36), Lead (E = 16 GPa, ฮฝ = 0.44) .
max
โข Loading: cyclic mechanical with Ux = 1 ยตm.
7. Cohesive zone method: cohesive zone?
continuum element
3 n 4
t โ cohesive zone
1 2
continuum element
โข Cohesive zones are embedded between continuum elements.
โข Constitutive behavior: speci๏ฌed through a relation between
the separation โ (initially = 0) and a corresponding traction T(โ).
8. Cohesive zone method: stiffness matrix and nodal force vector
โข The cohesive zone nodal displacement vector is constructed in the local frame
of reference (t,n):
uT = {u1, u1 , u2, u2 , u3, u3 , u4, u4 }.
t n t n t n t n
โข The relative displacement vector โ is then calculated as:
๏ฃฑ ๏ฃผ
๏ฃด
๏ฃฒ โ ๏ฃด
t ๏ฃฝ
โ=๏ฃด = Au
๏ฃณ โn ๏ฃด
๏ฃพ
where A is a matrix of the shape functions:
๏ฃฎ ๏ฃน
โh1 0 โh2 0 h1 0 h2 0
A=๏ฃฏ ๏ฃบ
0 โh1 0 โh2 0 h1 0 h2
๏ฃฐ ๏ฃป
and
1 1
h1 = (1 โ ฮท), h2 = (1 + ฮท).
2 2
The parameter ฮท is de๏ฌned at the cohesive zone mid plane and varies between
โ1 at nodes (1,3) and 1 at nodes (2,4).
9. โข The cohesive zone internal nodal force vector and stiffness matrix are now writ-
ten as:
l +1
f = S ATT dS = โ1 ATT dฮท
2
l +1
K = S ATBA dS = โ1 ATBA dฮท
2
where S is the cohesive zone area, l is the cohesive zone length and B is the
cohesive zone constitutive tangent operator given by:
โTt โTt ๏ฃบ
๏ฃฎ ๏ฃน
๏ฃฏ
โโt โโn ๏ฃบ
๏ฃฏ ๏ฃบ
๏ฃฏ
B= ๏ฃฏ
๏ฃบ.
๏ฃบ
โTn โTn ๏ฃบ
๏ฃฏ
๏ฃฏ
๏ฃฏ ๏ฃบ
๏ฃฐ ๏ฃป
โโt โโn
โข Finally, K and f are transformed to the global frame of reference (x,y).
10. Cohesive tractions: monotonic loading
1 1
Tn/ฯmax
0
Tt/ฯmax
0
โ1
โ2 (a) โ1 (b)
โ1 0 1 2 3 4 5 6 โ3 โ2 โ1 0 1 2 3
โ /ฮด โ /ฮด
n n t t
Cohesive zone monotonic normal (a) and shear (b) tractions.
โข Characteristics: peak traction and cohesive energy.
โข The softening branch is the energy dissipation source.
11. Cohesive tractions: cyclic loading
โข A linear relation is assumed between the cohesive traction and the corresponding
cohesive opening:
Tฮฑ = kฮฑ (1 โ Dฮฑ )โฮฑ
where kฮฑ is the initial stiffness and ฮฑ is either the local normal (n) or tangential
(t) direction in the cohesive zone plane.
โข Energy dissipation is accounted for by the damage variable D.
โข The damage variable is supplemented with an evolution law:
ห ห
D = f (โ, โ, T, D, ...).
12. Cyclic loading: damage evolution
โข Evolution law (motivated by Roe and Siegmund, 2003):
๏ฃซ ๏ฃถ
|Tฮฑ |
Dฮฑ = cฮฑ |โฮฑ | (1 โ Dฮฑ + r)m ๏ฃฌ
ห ห ๏ฃญ โ ฯf ๏ฃท
๏ฃธ
1 โ Dฮฑ
where cฮฑ , r, m are constants and ฯf is the cohesive zone endurance limit.
โข Satis๏ฌes main experimental observations on cyclic damage:
โ Damage increases with the number of cycles.
โ The larger the load, the larger the induced damage.
โ Damage is larger in the presence of mean stress/strain.
โ Load sequencing: cycling at a high stress level followed by a lower level
(HโL) causes more damage than when the order is reversed (LโH).
ฯf = 0 โโ linear damage accumulation (Minerโs law).
14. Initial cohesive stiffness
High initial stiffness โ minimize arti๏ฌcial enhancement of the overall compliance.
For a bar containing n equally spaced cohesive zones:
(U โ nโ)
ฯ= E,
L
T = k(1 โ D)โ.
Stress continuity โ ฯ = (U/L)E โ, where E โ is given as:
๏ฃซ ๏ฃถ
1
E โ = ๏ฃฌ1 โ kL
๏ฃญ ๏ฃธ E.
๏ฃท
nE (1 โ D) + 1
nE
To ensure a negligible enhancement of the overall compliance โ kL << 1.
E
In a two-dimensional model the condition is estimated by kl << 1, where l โ L/n
is the average cohesive zone length.
15. 400
0.15
(a)
0.1 (b) 200
T (MPa)
0.05
F (N)
0 0
โ0.05 โ200
โ0.1
โ0.15 โ400
0 200 400 600 800 1000 โ0.05 0 0.05 0.1 0.15 0.2
N (cycles) โ (ยต m)
(a) Reaction force vs. cycles to failure. (b) Cohesive traction vs. opening.
โข Assumption: damage does not occur under compression:
โ Physically: in๏ฌnite compressive strength.
โ Numerically: minimizes inter-penetration (overlapping) of neighboring con-
tinuum elements under compression.
16. F versus N : experimental (Erik de Kluizenaar: Philips).
17. 0.15 0.15
0.1 (a) 0.1 (b)
0.05 0.05
F (N)
F (N)
0 0
โ0.05 โ0.05
โ0.1 โ0.1
โ0.15 โ0.15
0 20 40 60 80 100 0 500 1000 1500 2000
N (cycles) N (cycles)
Different damage parameters: (a) r = 10โ3, m = 1. (b) r = 0, m = 3.
18. 1 1
HโL
0.8 0.8 LโH
0.6 ฮตmean = 0 0.6
D
ฮตmean = 0.5 %
D
0.4 0.4
0.2 (a)
0.2 (b)
0 0
0 200 400 600 800 1000 0 100 200 300 400
N (cycles) N (cycles)
(a) Mean strain effect. (b) Load sequencing effect.
HโL: 200 cycles at max = 1 % followd by 200 cycles at max = 0.5 %
LโH: 200 cycles at max = 0.5 % followd by 200 cycles at max = 1 %
19. Cohesive parameters: solder bump
czg1
czg2
czg3
czg4
โข Initial cohesive zone stiffness kฮฑ = 106 GPa/mm.
โ Suf๏ฌciently high compared to continuum stiffness. Identical for all cohesive
zone groups.
โข Damage coef๏ฌcient cฮฑ in [mm/N]: czg1 : 0, czg2 : 25, czg3 : 100, czg4 : 0.
โข ฯฮฑ = 0 MPa, r = 10โ3.
f
20. Computational time reduction
โข Loading is applied incrementally.
โข For large number of cycles โ time consuming.
โข Computational time reduction: only selected cycles are simulated.
โข Time reduction of more than 90 % in some cases.
21. Results: damage distribution
N = 500; Deff = 0.14 N = 1000; Deff = 0.22
Damage distribution in the solder bump at different cycles. Red lines indicate
i
damaged cohesive zones (De๏ฌ โฅ 0.5).
2 2
De๏ฌ = (Dn + Dt โ DnDt )1/2
i i i i i
23. 0.5
0.4
0.3
eff
D
0.2
0.1
0
0 2000 4000 6000 8000
N (cycles)
The total effective damage versus the number of cycles.
The total effective damage is calculated by averaging over all cohesive zones:
1 N
De๏ฌ = De๏ฌ S i
i
S i
i
where De๏ฌ is the effective damage at cohesive zone (i).
24. 8
6
4
F (N) 2
0
x
โ2
โ4
โ6
โ8
0 2000 4000 6000 8000
N (cycles)
The reaction force versus the number of cycles.
โข Slow softening followed by rapid softening (Kanchanomai et al., 2002)
25. S-N curve
โ0.5
FEM
) โ1 linear fit
max
โ1.5
log(ฮต
โ2
โ2.5
โ3
1 2 3 4 5 6
log(2N )
f
Applied strain max versus the number of reversals to failure 2Nf .
26. โข Finite element data can be ๏ฌtted with the Cof๏ฌn-Manson model:
max = a(2Nf )b
a: fatigue ductility coef๏ฌcient
b: fatigue ductility exponent
โข Failure criteria: 50% reduction in the reaction force
โโ a = 0.83, b = โ0.49.
โข Reduction of 25% or 75% โ same value of b.
โข Change by ยฑ50 % in the Youngโs modulii โ same value of b.
27. Effect of the elastic parameters
4
3
r
Nf/Nf
2
1
0
0.5 0.75 1 1.25 1.5
E/Er
Variation of Nf with E at max = 1%. Fitting curve: Nf /Nfr = (E/E r)โ1.83.
28. Conclusions
โข Evolution law captures main cyclic damage characteristics.
โข The modelโs prediction of the solder bump life-time agrees with the Cof๏ฌn-
Manson model.
โข More ef๏ฌcient computational time reduction scheme:
โโ simulation of larger number of cycles.
โโ more realistic microstructure.